Asymptotic vanishing conjecture for syzygies of large-degree embeddings

Let $X$ be a smooth projective variety of dimension $n$, let $A$ be an ample line bundle, let $P$ be a line bundle, and set

$$L_d=dA+P.$$

For a line bundle $B$ on $X$, write $K_{p,q}(X,B;L_d)$ for the Koszul cohomology group of weight $q$ and homological degree $p$ associated to the embedding defined by $L_d$.

Asymptotic vanishing conjecture. Fix $q\in[2,n]$. In the situation of the nonvanishing theorem referenced in the source, there is a constant $C_3$, depending on $X$, $A$, $B$ and $P$, such that

$$K_{p,q}(X,B;L_d)=0$$

for $p\leq C_3d^{q-1}$ when $d\gg0$.

The conjecture predicts that the lower bound in the corresponding asymptotic nonvanishing theorem is sharp: all smaller homological degrees eventually have vanishing syzygies. The source identifies this as the main open problem concerning the rough asymptotics of the groups $K_{p,q}$ for large $d$.